Numerical semigroups, polyhedra, and posets I: the group cone

TL;DR

This paper introduces group cones, a new family of polyhedra linked to finite abelian groups, providing a unified combinatorial framework to study rational polyhedra related to numerical semigroups.

Contribution

It defines group cones and shows their faces are indexed by finite posets, connecting them to existing polyhedral families associated with numerical semigroups.

Findings

Faces of group cones are indexed by finite posets

Group cones encompass families related to numerical semigroups

Provides a natural polyhedral geometric perspective

Abstract

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.